We perform a geometric analysis focusing on relaxation oscillations and canard cycles within a singularly perturbed predator-prey system involving budworm and birds. The system undergoes a comprehensive stability analysis, leading to the identification of canard cycles in proximity to the Hopf bifurcation points. The study particularly highlights the transition from smaller Hopf-type cycles to larger relaxation cycles. And the expression of transition threshold $\mu_c(\sqrt{ε})$ of the spruce budworm-bird system is obtained innovatively. Furthermore, numerical simulations are carried out to validate the theoretical findings.

Our focus in this study revolves around investigating the following parabolic problem

By using the Galerkin approximation and a family of potential wells, we obtain the existence of global solution and finite time blow-up under some suitable conditions. On the other hand, the results for asymptotic behavior of certain solutions with positive initial energy are also given.

This paper is concerned with a generalization of the modified Camassa-Holm equation with both cubic and quadratic nonlinearities (also known as the CH-mCH equation). We mainly prove the orbital stability of the train of peakons for the CH-mCH equation in energy space, using energy arguments and combining the method of orbital stability of a single peakon with the monotonicity of the local energy norm.

The focus of this study is on the existence of three solutions to ($p(·)$, $q(·)$)-biharmonic operator with an $s(x)$-Hardy term under no-flux boundary conditions. Our method is based on the variational method and critical point results.

A co-infection of a human or a pig with human influenza or COVID-19 strains and H5N1 strain may result in a pandemic strain, causing a widespread deadly pandemic. In this paper, we consider a new class of co-infections disease epidemic models for a rapid and slow virus. We study the transmission threshold by analyzing the basic reproduction number. The equilibrium points for the model are derived, and their local stability is analyzed with suitable assumptions on the model parameters. Understanding the model parameters is one of the prime subjects in this research work. Therefore, the sensitivity of essential parameters is investigated. Moreover, the optimal control problem for the proposed model is considered, and first-order optimality conditions are derived. Finally, numerical simulations indicate the effects of the model’s basic reproduction number and control variables.

For an anisotropic discrete nonlinear problem with a variable exponent, we demonstrate both the existence and multiplicity of nontrivial solutions in this study. Our technique is based on variational methods.

The objective of this article is to present a new class of vector interval-valued variational-like inequality problems. Based on the concepts of LU optimal and weakly LU optimal solutions, we derive some relations between the interval-valued programming and variational-like inequality problems. The study of the interplay between interval-valued optimization problems (IVOP) and vector variational-like inequalities (VVLI) combines theoretical advancements under the concept of differentiability and $\mu$- invexity. Furthermore, to demonstrate the established linkages, we provide examples and an example demonstrates how well the vector variational inequality problems may be applied to deal with (MOIVP) problems.

For the large sparse complex symmetric linear systems, by applying the minimal residual technique to accelerate a preconditioned variant of new Hermitian and skew-Hermitian splitting (${\rm P}^∗{\rm NHSS}$) method and efficient parameterized ${\rm P}^∗{\rm NHSS}$ $({\rm PPNHSS})$ method, we construct the minimal residual ${\rm P}^∗{\rm NHSS}$ $({\rm MRP}^∗{\rm NHSS})$ method and the minimal residual ${\rm PPNHSS}$ $({\rm MRPPNHSS})$ method. The convergence properties of the two iteration methods are studied. Theoretical analyses imply that the ${\rm MRP}^∗{\rm NHSS}$ method and the ${\rm MRPPNHSS}$ method converge unconditionally to the unique solution. In addition, we also give the inexact versions of ${\rm MRP}^∗{\rm NHSS}$ method and ${\rm MRPPNHSS}$ method and their convergence proofs. Finally, numerical experiments show the high efficiency and robustness of our methods.

In this paper, we are interested in proving a general fixed point theorem for multivalued mappings in fuzzy $b$−metric spaces. The results presented in this paper not only generalize the findings from [23], but also yield additional specific outcomes. We present an application to establish the existence of a solution to the integral equation, demonstrating the significance of our result.

We consider a first-order nonlinear neutral differential equation. By employing Krasnoselskii’s fixed point theorem, we provide several new criteria for the existence of positive periodic solutions to this equation. The theorems we have formulated are exemplified through a specific example.

This study presents an advanced delayed spatiotemporal epidemiological model that incorporates a Holling type II incidence rate to capture the saturation effects observed in disease transmission dynamics during the COVID-19 pandemic. The model integrates two crucial intervention measures - vaccination of susceptible individuals and hospitalization of severe cases - while accounting for both spatial diffusion and the latent period within the epidemic compartments. This framework facilitates the precise optimization of vaccination and hospitalization strategies as functions of spatial location and temporal evolution, yielding new insights into spatially targeted public health interventions. We rigorously analyze the model equilibrium points, establishing conditions for their existence and local stability. An optimal control problem is formulated, uniquely considering the combined effects of spatial diffusion and latent period, with controls dynamically varying across space and time. The well-posedness of the control problem is verified, supported by proofs of existence, uniqueness, positivity, and boundedness of the strong solution. First-order necessary optimality conditions are derived, characterizing the optimal vaccination and hospitalization strategies through state and adjoint variables. Numerical simulations across diverse intervention scenarios demonstrate the effectiveness of adaptive, space-time-specific control strategies in mitigating COVID-19 transmission. This work offers a novel mathematical and computational approach to the optimal spatiotemporal management of epidemic control measures.

A novel coronavirus (COVID-19) causes an outbreak of viral pneumonia in Wuhan, Hubei province, China. In this paper, the deterministic compartmental model is given based on the clinical progression of the disease and the intervention measures implemented by the Chinese authorities. Simulations of the model are given to estimate the basic reproduction number for COVID-19 based on the daily reported cases from China CDC. The basic reproduction number of the model is used to assess the transmissibility of COVID-19. The results indicate that COVID-19 will be controlled at the end of March if there are no imported infections people into China. Then the first-level public health emergency response can be adjusted to the second-level or the higher-level response.

The HIV/AIDS epidemic profoundly affects the working-class population, resulting in increased mortality and morbidity rates and causing substantial labour loss across various sectors. To assess the epidemic’s impact on the specified class, a fractal-fractional-order mathematical model was formulated based on Atangana-Baleanu-Caputo operator. This model encompasses seven compartments, and the existence and uniqueness of its solutions were verified based on the Banach fixed-point theorem, contraction mapping concepts, and Hyers-Ulam stability criteria. The mathematical model was analyzed to understand the effects of HIV/AIDS on the working-class population. Real data from Ethiopia were utilized to validate the model. Subsequently, the model was extended to optimal control fractal-fractional models, incorporating different control strategies. Numerical simulations were performed using MATLAB R2019a to support the analytical solutions. The study’s results demonstrated that the fractal-fractional-order model provides a comprehensive understanding of the complexities of HIV/AIDS infection. The findings suggested that increasing the number of infected productive members of the population can help to control the spread of the disease by reducing the inequality caused by HIV/AIDS. Furthermore, the numerical simulation of the optimal control model indicated that decreasing non-productivity reduces the number of infected individuals. Therefore, effective management of the infection holds the potential to eradicate the disease in the country.

In this paper, we consider the three-dimensional generalized Navier-Stokes equation with a nonlinear damping term $|u|^{β−1} (β ≥ 1).$ Firstly, utilizing the Fourier splitting method, we derive decay estimates for weak solutions to the equations when $α = 0$ and $β = 1,$ as well as when $0 < α <\frac{3}{4}$ for any $β = 2.$ Secondly, for $0 < α < \frac{5}{4}$ and any $β > {\rm max}\{ \frac{4α}{3}+1, 2\},$ we obtain the same result.

In this article, the fixed point of the SM iterative approach is approximated via Suzuki mapping. In the context of Banach spaces, we provide both weak and strong convergence for the SM iteration. Then, by comparing it with certain well-known iterations, we give some numerical examples to show the effectiveness of SM iteration for the Suzuki-type mapping.

This article presents a direct method for calculating the normal form coefficients of a 1:1 resonant Hopf bifurcation in reaction-diffusion systems with time delay and Neumann boundary conditions. The formulas obtained in this paper can be easily implemented using a computer algebra system such as Maple or Mathematica.

In this paper, we establish the local existence for the generalized Navier-Stokes-Maxwell system with the fractional velocity dissipative term $Λ^{2α}u$ and fractional magnetic dissipative term $Λ^{2β}B.$ Moreover, we establish the global existence of strong solutions to this generalized model.

In ecology, it is of great significance to research the influence of gestation period on the dynamics of eco-epidemiology. In the paper, we establish and explore a delayed predator-pest model with disease in pest. We first analyze the existence and local stability of each equilibrium of the model. Then, we investigate the existence of Hopf bifurcation at the coexistence equilibrium. Moreover, we calculate the normal form to examine the properties of Hopf bifurcation. Some numerical simulations are conducted to verify the theoretical results obtained and explore how the delay affects the biomass of pest. Our findings may contribute to a better understanding of the mechanisms of interaction between species in eco-epidemiology. At the same time, this study also provides an insightful perspective into the control of pests in ecosystems.

For some $abcd$-water wave models describing small amplitude, long wavelength gravity waves on the surface of water, in this paper, by using the method of dynamical systems to analyze corresponding traveling wave systems and find the bifurcations of phase portraits, the dynamical behavior of systems can be derived. Under some given parameter conditions, for a wave component, the existence of periodic wave solutions, solitary wave solutions, kink and anti-kink wave solutions as well as compacton families can be proved. Possible exact explicit parametric representations of the traveling wave solutions are given.